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EIILM UNIVERSITY, SIKKIM EXAMINATIONS, FEBRUARY -2012 B.A.ADD. (MATHEMATICS), YEAR-1 ALGEBRA Time: 3 hours M.Marks:60 Note: - Attempt any 5 questions. All questions carry equal marks. Q.1 Explain Vector spaces and dimension and coordinates of a vector space? A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, calledscalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. An example of a vector space is that of Euclidean vectors, which may be used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vector spaces are the subject of linear algebra and are well understood from this point of view, since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm orinner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory. Historically, the first ideas leading to vector spaces can be traced back as far as 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in the late 19th century, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs. Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework forFourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. First example: arrows in the plane The concept of vector space will first be explained by describing two particular examples. The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w. Another operation that can be done with arrows is scaling: given any positive real number a, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction, instead. The following shows a few examples: if a = 2, the resulting vector aw has the same direction as w, but is stretched to the double length of w (right image below). Equivalently 2w is the sum w + w. Moreover, (−1)v = −v has the opposite direction and the same length as v (blue vector pointing down in the right image). [edit]Second example: ordered pairs of numbers A second key example of a vector space is provided by pairs of real numbers x and y. (The order of the components x and y is significant, so such a pair is also called an ordered pair.) Such a pair is written as (x, y). The sum of two such pairs and multiplication of a pair with a number is defined as follows: (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) and a (x, y) = (ax, ay). [edit]Definition A vector space over a field F is a set V together with two binary operations that satisfy the eight axioms listed below. Elements of V are called vectors. Elements of F are called scalars. In this article, vectors are distinguished from scalars by boldface. [nb 1] In the two examples above, our set consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively, while our field is the real numbers. The first operation, vector addition, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. The second operation takes any scalar a and any vector v and gives another vector av. In view of the first example, where the multiplication is done by rescaling the vector v by a scalar a, the multiplication is called scalar multiplication of v by a. Q.2 Explain representation of transformations by matrices? An m×n matrix is a set of numbers arranged in m rows and n columns. The following illustration shows several matrices. You can add two matrices of the same size by adding individual elements. The following illustration shows two examples of matrix addition. An m×n matrix can be multiplied by an n×p matrix, and the result is an m×p matrix. The number of columns in the first matrix must be the same as the number of rows in the second matrix. For example, a 4 ×2 matrix can be multiplied by a 2 ×3 matrix to produce a 4 ×3 matrix. Points in the plane and rows and columns of a matrix can be thought of as vectors. For example, (2, 5) is a vector with two components, and (3, 7, 1) is a vector with three components. The dot product of two vectors is defined as follows: (a, b) • (c, d) = ac + bd (a, b, c) • (d, e, f) = ad + be + cf For example, the dot product of (2, 3) and (5, 4) is (2)(5) + (3)(4) = 22. The dot product of (2, 5, 1) and (4, 3, 1) is (2)(4) + (5)(3) + (1)(1) = 24. Note that the dot product of two vectors is a number, not another vector. Also note that you can calculate the dot product only if the two vectors have the same number of components. Let A(i, j) be the entry in matrix A in the ith row and the jth column. For example A(3, 2) is the entry in matrix A in the 3rd row and the 2nd column. Suppose A, B, and C are matrices, and AB = C. The entries of C are calculated as follows: C(i, j) = (row i of A) • (column j of B) The following illustration shows several examples of matrix multiplication. If you think of a point in the plane as a 1 × 2 matrix, you can transform that point by multiplying it by a 2 × 2 matrix. The following illustration shows several transformations applied to the point (2, 1). All the transformations shown in the previous figure are linear transformations. Certain other transformations, such as translation, are not linear, and cannot be expressed as multiplication by a 2 × 2 matrix. Suppose you want to start with the point (2, 1), rotate it 90 degrees, translate it 3 units in the x direction, and translate it 4 units in the y direction. You can accomplish this by performing a matrix multiplication followed by a matrix addition. A linear transformation (multiplication by a 2 × 2 matrix) followed by a translation (addition of a 1 × 2 matrix) is called an affine transformation. An alternative to storing an affine transformation in a pair of matrices (one for the linear part and one for the translation) is to store the entire transformation in a 3 × 3 matrix. To make this work, a point in the plane must be stored in a 1 × 3 matrix with a dummy 3rd coordinate. The usual technique is to make all 3rd coordinates equal to 1. For example, the point (2, 1) is represented by the matrix [2 1 1]. The following illustration shows an affine transformation (rotate 90 degrees; translate 3 units in the x direction, 4 units in the y direction) expressed as multiplication by a single 3 × 3 matrix. In the previous example, the point (2, 1) is mapped to the point (2, 6). Note that the third column of the 3 × 3 matrix contains the numbers 0, 0, 1. This will always be the case for the 3 × 3 matrix of an affine transformation. The important numbers are the six numbers in columns 1 and 2. The upper-left 2 × 2 portion of the matrix represents the linear part of the transformation, and the first two entries in the 3rd row represent the translation. In Windows GDI+ you can store an affine transformation in a Matrix object. Because the third column of a matrix that represents an affine transformation is always (0, 0, 1), you specify only the six numbers in the first two columns when you construct a Matrix object. The statement Matrix myMatrix(0.0f, 1.0f, -1.0f, 0.0f, 3.0f, 4.0f); constructs the matrix shown in the previous figure. Q.3 Explain the transpose of a linear transformation with suitable examples? Q.4 Explain characteristics values and Characteristics vectors? Diagonalizable operators, Q.5 Derive primary decomposition theorem? Let T be a linear operater on the finite dimensional vector space V over the field F. Let be the factorization of mT into irreducible monic polynomials over F. Let 1. . Then ; 2. each Wi is T-invariant; 3. if Ti is the operator induced on Wi by T, then mTi=piri. Example 6.7.1 Let operator on V=F3. Then or and , considered as a linear and . Thus A is not diagonalizable over F. The A-invariant docomposition of V is where and . Now and its kernel is spanned by (0,2,1) and (1,2,1). Note that (1,2,1) is an eigenvector. The kernel of A-1 is generated by (0,1,1), which is an eigenvector. Since A(0,2,1)t=-2(1,2,1)t+1(0,2,1)t, we see that with respect to the ordered basis , A can be put in the form . Now consider the matrix . We have mA=(x-1)2. Thus the primary docomposition theorem does not give any decomposition. Nevertheless, it is obvious that V=F3 can be decomposed into 1-dimensional and 2dimensional A-invariant subspaces. Suppose that the minimal polynomial is a product of linear factors. As in the proof of the above theorem, let Ei=fi(T)gi(T), basis . Choose a for each Wi and collect them to get a basis Then of V. . If we let then the diagonalizable part of T. Now let . So D is diagonalizable. D is called Then since EiEj=0 if and Ei2=Ei. Thus for r>ri for all i, we have Nr=0. Since D and N are polynomials in T, we have DN=ND. Definition 6.15 If N is a operator on a n-dim vector space V such that Nr=0 for some posivive integer r, then N is called a nilpotent operator. Note that if N is nilpotent, then Nr=0 for some . Q.6 Derive the spectral theorem on operators? Q.7 Explain simultaneous diagonalization of normal operators? In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix. If V is a finitedimensional vector space, then a linear map T : V → V is called diagonalizable if there exists a basis of V with respect to which T is represented by a diagonal matrix. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. [1] A square matrix which is not diagonalizable is called defective. Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle: their eigenvalues and eigenvectors are known and one can raise a diagonal matrix to a power by simply raising the diagonal entries to that same power. Geometrically, a diagonalizable matrix is an inhomogeneous dilation (or anisotropic scaling) – it scales the space, as does ahomogeneous dilation, but by a different factor in each direction, determined by the scale factors on each axis (diagonal entries). The fundamental fact about diagonalizable maps and matrices is expressed by the following: An n-by-n matrix A over the field F is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to n, which is the case if and only if there exists a basis of Fn consisting of eigenvectors of A. If such a basis has been found, one can form the matrix P having these basis vectors as columns, and P−1AP will be a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of A. A linear map T: V → V is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to dim(V), which is the case if and only if there exists a basis of V consisting of eigenvectors of T. With respect to such a basis, T will be represented by a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of T. Another characterization: A matrix or linear map is diagonalizable over the field F if and only if its minimal polynomial is a product of distinct linear factors over F. (Put in another way, a matrix is diagonalizable if and only if all of its elementary divisors are linear.) The following sufficient (but not necessary) condition is often useful. An n-by-n matrix A is diagonalizable over the field F if it has n distinct eigenvalues in F, i.e. if its characteristic polynomial has n distinct roots in F; however, the converse may be false. Let us consider which has eigenvalues 1, 2, 2 (not all distinct) and is diagonalizable with diagonal form (also the similar matrix of A) A linear map T: V → V with n = dim(V) is diagonalizable if it has n distinct eigenvalues, i.e. if its characteristic polynomial has n distinct roots in F. Let A be a matrix over F. If A is diagonalizable, then so is any power of it. Conversely, if A is invertible, F is algebraically closed, and An is diagonalizable for some n that is not an integer multiple of the characteristic of F, then A is diagonalizable. Proof: If An is diagonalizable, then A is annihilated by some polynomial (since , which has no multiple root ) and is divided by the minimal polynomial of A. As a rule of thumb, over C almost every matrix is diagonalizable. More precisely: the set of complex n-by-n matrices that are not diagonalizable over C, considered as a subset of Cn×n, has Lebesgue measure zero. One can also say that the diagonalizable matrices form a dense subset with respect to the Zariski topology: the complement lies inside the set where the discriminant of the characteristic polynomial vanishes, which is a hypersurface. From that follows also density in the usual (strong) topology given by a norm. The same is not true over R. The Jordan–Chevalley decomposition expresses an operator as the sum of its semisimple (i.e., diagonalizable) part and its nilpotent part. Hence, a matrix is diagonalizable if and only if its nilpotent part is zero. Put in another way, a matrix is diagonalizable if each block in its Jordan form has no nilpotent part; i.e., one-by-one matrix. Q.8 Write differences between symmetric bilinear forms and Skew Symmetric bilinear forms?